3.4. Properties of Perpendicular Lines http://www.ck12.org
Given:l||m,l⊥n
Prove:n⊥m
TABLE3.9:
Statement Reason
1.l||m,l⊥n Given2.^61 ,^62 ,^6 3, and^6 4 are right angles Definition of perpendicular lines
3.m^61 = 90 ◦ Definition of a right angle
4.m^61 =m^65 Corresponding Angles Postulate
5.m^65 = 90 ◦ Transitive PoE
6.m^66 =m^67 = 90 ◦ Congruent Linear Pairs
7.m^68 = 90 ◦ Vertical Angles Theorem
8.^65 ,^66 ,^6 7, and^6 8 are right angles Definition of right angle
9.n⊥m Definition of perpendicular lines
Theorem 3-1: If two lines are parallel and a third line is perpendicular to one of the parallel lines, it is also
perpendicular to the other parallel line.
Or, ifl||mandl⊥n, thenn⊥m.
Theorem 3-2:If two lines are perpendicular to the same line, they are parallel to each other.
Or, ifl⊥nandn⊥m, thenl||m. You will prove this theorem in the review questions.
From these two theorems, we can now assume that any angle formed by two parallel lines and a perpendicular
transversal will always be 90◦.
Example 2:Determine the measure of^6 1.
Solution:From Theorem 3-1, we know that the lower parallel line is also perpendicular to the transversal. Therefore,
m^61 = 90 ◦.
Adjacent Complementary Angles
Recall that complementary angles add up to 90◦. If complementary angles are adjacent, their nonadjacent sides are
perpendicular rays. What you have learned about perpendicular lines can be applied to this situation.
Example 3:Findm^6 1.
Solution:The two adjacent angles add up to 90◦, sol⊥m. Therefore,m^61 = 90 ◦.